INSIGHTS Workshop on Statistics and Machine Learning CERN, 17-21 September, 2018

1. Statistics for Particle PhysicsLecture 1: Fundamentals https://indico.cern.ch/event/747653/ INSIGHTS Workshop on Statistics and Machine Learning CERN, 17-21 September, 2018 Glen Cowan Physics Department Royal Holloway, University of London g.cowan@rhul.ac.uk www.pp.rhul.ac.uk/~cowan CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

2. Outline Lecture1: Fundamentals Probability Random variables, pdfs Lecture 2: Statistical tests Formalism of frequentist tests Comments on multivariate methods (brief) p-values Discovery and limits Lecture 3: Parameter estimation Properties of estimators Maximum likelihood CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

3. Some statistics books, papers, etc. G. Cowan, Statistical Data Analysis, Clarendon, Oxford, 1998 R.J. Barlow, Statistics: A Guide to the Use of Statistical Methods in the Physical Sciences, Wiley, 1989 Ilya Narsky and Frank C. Porter, Statistical Analysis Techniques in Particle Physics, Wiley, 2014. Luca Lista, Statistical Methods for Data Analysis in Particle Physics, Springer, 2017. L. Lyons, Statistics for Nuclear and Particle Physics, CUP, 1986 F. James., Statistical and Computational Methods in Experimental Physics, 2nd ed., World Scientific, 2006 S. Brandt, Statistical and Computational Methods in Data Analysis, Springer, New York, 1998 (with program library on CD) M. Tanabashi et al. (PDG), Phys. Rev. D 98, 030001 (2018); see also pdg.lbl.gov sections on probability, statistics, Monte Carlo CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

4. Theory ↔ Statistics ↔ Experiment Theory (model, hypothesis): Experiment: + data selection + simulation of detector and cuts CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

5. Data analysis in particle physics Observe events (e.g., pp collisions) and for each, measure a set of characteristics: particle momenta, number of muons, energy of jets,... Compare observed distributions of these characteristics to predictions of theory. From this, we want to: Estimate the free parameters of the theory: Quantify the uncertainty in the estimates: Assess how well a given theory stands in agreement with the observed data: To do this we need a clear definition of PROBABILITY CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

6. A definition of probability Consider a set S with subsets A, B, ... Kolmogorov axioms (1933) Also define conditional probability of A given B: Subsets A, Bindependent if: If A, B independent, CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

7. Interpretation of probability I. Relative frequency A, B, ... are outcomes of a repeatable experiment cf. quantum mechanics, particle scattering, radioactive decay... II. Subjective probability A, B, ... are hypotheses (statements that are true or false) • Both interpretations consistent with Kolmogorov axioms. • In particle physics frequency interpretation often most useful, but subjective probability can provide more natural treatment of non-repeatable phenomena: systematic uncertainties, probability that Higgs boson exists,... CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

8. Bayes’ theorem From the definition of conditional probability we have, and , so but Bayes’ theorem First published (posthumously) by the Reverend Thomas Bayes (1702−1761) An essay towards solving a problem in the doctrine of chances, Philos. Trans. R. Soc. 53 (1763) 370; reprinted in Biometrika, 45 (1958) 293. CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

9. B The law of total probability Consider a subset B of the sample space S, S divided into disjoint subsets Ai such that ∪i Ai = S, Ai B∩ Ai → → law of total probability → Bayes’ theorem becomes CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

10. An example using Bayes’ theorem Suppose the probability (for anyone) to have a disease D is: ←prior probabilities, i.e., before any test carried out Consider a test for the disease: result is + or - ←probabilities to (in)correctly identify a person with the disease ←probabilities to (in)correctly identify a healthy person Suppose your result is +. How worried should you be? CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

11. Bayes’ theorem example (cont.) The probability to have the disease given a + result is ← posterior probability i.e. you’re probably OK! Your viewpoint: my degree of belief that I have the disease is 3.2%. Your doctor’s viewpoint: 3.2% of people like this have the disease. CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

12. Frequentist Statistics − general philosophy In frequentist statistics, probabilities are associated only with the data, i.e., outcomes of repeatable observations (shorthand: ). Probability = limiting frequency Probabilities such as P (Higgs boson exists), P (0.117 < αs < 0.121), etc. are either 0 or 1, but we don’t know which. The tools of frequentist statistics tell us what to expect, under the assumption of certain probabilities, about hypothetical repeated observations. A hypothesis is is preferred if the data are found in a region of high predicted probability (i.e., where an alternative hypothesis predicts lower probability). CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

13. Bayesian Statistics − general philosophy In Bayesian statistics, use subjective probability for hypotheses: probability of the data assuming hypothesis H (the likelihood) prior probability, i.e., before seeing the data posterior probability, i.e., after seeing the data normalization involves sum over all possible hypotheses Bayes’ theorem has an “if-then” character: If your prior probabilities were π(H), then it says how these probabilities should change in the light of the data. No general prescription for priors (subjective!) CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

14. Random variables and probability density functions A random variable is a numerical characteristic assigned to an element of the sample space; can be discrete or continuous. Suppose outcome of experiment is continuous value x →f (x) = probability density function (pdf) x must be somewhere Or for discrete outcome xi with e.g. i = 1, 2, ... we have probability mass function x must take on one of its possible values CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

15. Cumulative distribution function Probability to have outcome less than or equal to x is cumulative distribution function Alternatively define pdf with CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

16. Other types of probability densities Outcome of experiment characterized by several values, e.g. an n-component vector, (x1, ... xn) →joint pdf Sometimes we want only pdf of some (or one) of the components →marginal pdf x1, x2 independent if Sometimes we want to consider some components as constant →conditional pdf CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

17. Functions of a random variable A function of a random variable is itself a random variable. Suppose x follows a pdf f(x), consider a function a(x). What is the pdf g(a)? dS = region of x space for which a is in [a, a+da]. For one-variable case with unique inverse this is simply → CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

18. Functions without unique inverse If inverse of a(x) not unique, include all dx intervals in dS which correspond to da: Example: CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

19. Functions of more than one r.v. and a function Consider r.v.s dS = region of x-space between (hyper)surfaces defined by CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

20. Functions of more than one r.v. (2) Example: r.v.s x, y > 0 follow joint pdf f(x,y), consider the function z = xy. What is g(z)? → (Mellin convolution) CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

21. More on transformation of variables Consider a random vector with joint pdf Form n linearly independent functions for which the inverse functions exist. Then the joint pdf of the vector of functions is where J is the Jacobian determinant: For e.g. integrate over the unwanted components. CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

22. Expectation values Consider continuous r.v. x with pdf f (x). Define expectation (mean) value as Notation (often): ~ “centre of gravity” of pdf. For a function y(x) with pdf g(y), (equivalent) Variance: Notation: Standard deviation: σ ~ width of pdf, same units as x. CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

23. Covariance and correlation Define covariance cov[x,y] (also use matrix notation Vxy) as Correlation coefficient (dimensionless) defined as If x, y, independent, i.e., , then → x and y, ‘uncorrelated’ N.B. converse not always true. CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

24. Correlation (cont.) CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

25. Error propagation Suppose we measure a set of values and we have the covariances which quantify the measurement errors in the xi. Now consider a function What is the variance of to find the pdf The hard way: use joint pdf then from g(y) find V[y] = E[y2] - (E[y])2. may not even be fully known. Often not practical, CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

26. Error propagation (2) Suppose we had in practice only estimates given by the measured Expand to 1st order in a Taylor series about To find V[y] we need E[y2] and E[y]. since CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

27. Error propagation (3) Putting the ingredients together gives the variance of CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

28. Error propagation (4) If the xi are uncorrelated, i.e., then this becomes Similar for a set of m functions or in matrix notation where CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

29. Error propagation (5) y(x) The ‘error propagation’ formulae tell us the covariances of a set of functions in terms of the covariances of the original variables. σy x σx Limitations: exact only if linear. y(x) Approximation breaks down if function nonlinear over a region comparable in size to the σi. ? x σx N.B. We have said nothing about the exact pdf of the xi, e.g., it doesn’t have to be Gaussian. CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

30. Error propagation − special cases → → That is, if the xi are uncorrelated: add errors quadratically for the sum (or difference), add relative errors quadratically for product (or ratio). But correlations can change this completely... CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

31. Error propagation − special cases (2) Consider with Now suppose ρ = 1. Then i.e. for 100% correlation, error in difference → 0. CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

32. Some distributions Distribution/pdf Example use in HEP Binomial Branching ratio Multinomial Histogram with fixed N Poisson Number of events found Uniform Monte Carlo method Exponential Decay time Gaussian Measurement error Chi-square Goodness-of-fit Cauchy Mass of resonance Landau Ionization energy loss Beta Prior pdf for efficiency Gamma Sum of exponential variables Student’s t Resolution function with adjustable tails CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

33. Binomial distribution Consider N independent experiments (Bernoulli trials): outcome of each is ‘success’ or ‘failure’, probability of success on any given trial is p. Define discrete r.v. n = number of successes (0 ≤ n ≤ N). Probability of a specific outcome (in order), e.g. ‘ssfsf’ is But order not important; there are ways (permutations) to get n successes in N trials, total probability for n is sum of probabilities for each permutation. CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

34. Binomial distribution (2) The binomial distribution is therefore parameters random variable For the expectation value and variance we find: CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

35. Binomial distribution (3) Binomial distribution for several values of the parameters: Example: observe N decays of W±, the number n of which are W→μν is a binomial r.v., p = branching ratio. CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

36. Multinomial distribution Like binomial but now m outcomes instead of two, probabilities are For N trials we want the probability to obtain: n1 of outcome 1, n2 of outcome 2, ⠇ nm of outcome m. This is the multinomial distribution for CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

37. Multinomial distribution (2) Now consider outcome i as ‘success’, all others as ‘failure’. → all ni individually binomial with parameters N, pi for all i One can also find the covariance to be represents a histogram Example: with m bins, N total entries, all entries independent. CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

38. Poisson distribution Consider binomial n in the limit → n follows the Poisson distribution: Example: number of scattering events n with cross section σ found for a fixed integrated luminosity, with CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

39. Uniform distribution Consider a continuous r.v. x with -∞ < x < ∞ . Uniform pdf is: N.B. For any r.v. x with cumulative distribution F(x), y = F(x) is uniform in [0,1]. Example: for π0→ γγ, Eγ is uniform in [Emin, Emax], with CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

40. Exponential distribution The exponential pdf for the continuous r.v. x is defined by: Example: proper decay time t of an unstable particle (τ = mean lifetime) Lack of memory (unique to exponential): CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

41. Gaussian distribution The Gaussian (normal) pdf for a continuous r.v. x is defined by: (N.B. often μ, σ2 denote mean, variance of any r.v., not only Gaussian.) Special case: μ = 0, σ2 = 1 (‘standard Gaussian’): If y ~ Gaussian with μ, σ2, then x = (y-μ) /σ follows φ(x). CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

42. Gaussian pdf and the Central Limit Theorem The Gaussian pdf is so useful because almost any random variable that is a sum of a large number of small contributions follows it. This follows from the Central Limit Theorem: For n independent r.v.s xi with finite variances σi2, otherwise arbitrary pdfs, consider the sum In the limit n→ ∞, y is a Gaussian r.v. with Measurement errors are often the sum of many contributions, so frequently measured values can be treated as Gaussian r.v.s. CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

43. Central Limit Theorem (2) The CLT can be proved using characteristic functions (Fourier transforms), see, e.g., SDA Chapter 10. For finite n, the theorem is approximately valid to the extent that the fluctuation of the sum is not dominated by one (or few) terms. Beware of measurement errors with non-Gaussian tails. Good example: velocity component vx of air molecules. OK example: total deflection due to multiple Coulomb scattering. (Rare large angle deflections give non-Gaussian tail.) Bad example: energy loss of charged particle traversing thin gas layer. (Rare collisions make up large fraction of energy loss, cf. Landau pdf.) CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

44. Multivariate Gaussian distribution Multivariate Gaussian pdf for the vector are transpose (row) vectors, are column vectors, For n = 2 this is where ρ = cov[x1, x2]/(σ1σ2) is the correlation coefficient. CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

45. Chi-square (χ2) distribution The chi-square pdf for the continuous r.v. z (z≥ 0) is defined by n = 1, 2, ... = number of ‘degrees of freedom’ (dof) For independent Gaussian xi, i = 1, ..., n, means μi, variances σi2, follows χ2 pdf with n dof. Example: goodness-of-fit test variable especially in conjunction with method of least squares. CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

46. Cauchy (Breit-Wigner) distribution The Breit-Wigner pdf for the continuous r.v. x is defined by (Γ = 2, x0 = 0 is the Cauchy pdf.) E[x] not well defined, V[x] →∞. x0 = mode (most probable value) Γ = full width at half maximum Example: mass of resonance particle, e.g. ρ, K*, φ0, ... Γ = decay rate (inverse of mean lifetime) CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

47. Landau distribution For a charged particle with β = v /c traversing a layer of matter of thickness d, the energy loss Δ follows the Landau pdf: Δ + - + - β - + - + d L. Landau, J. Phys. USSR 8 (1944) 201; see also W. Allison and J. Cobb, Ann. Rev. Nucl. Part. Sci. 30 (1980) 253. CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

48. Landau distribution (2) Long ‘Landau tail’ →all moments ∞ Mode (most probable value) sensitive to β, →particle i.d. CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

49. Beta distribution Often used to represent pdf of continuous r.v. nonzero only between finite limits. CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1

50. Gamma distribution Often used to represent pdf of continuous r.v. nonzero only in [0,∞]. Also e.g. sum of n exponential r.v.s or time until nth event in Poisson process ~ Gamma CERN, INSIGHTS Statistics Workshop / 17-21 Sep 2018 / Lecture 1